Method and apparatus for controlling thickness of elongated workpieces



April 1967 HANS'FRIEDRICH MARTEN 3,

METHOD AND APPARATUS FOR CONTROLLING THICKNESS OF ELONGATED WORKPIECES Filed Dec. 20, 1963 6 Sheets-Sheet 1 INVENTOR.

HANS F. MARTEN HIS ATTORNEY April 25, 1967 HANS-FRIEDRICH MARTEN 3,3

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momom 4401 m 33255 505 5 16m @zExE wzimio m3 om A Qk 3980:! TIOH United States Patent S 11 Claims. (Cl. 72-16) This invention relates to a method of and apparatus for the production of strip, bars or the like of constant thickness, especially those of steel or other metal by rolling down starting material of irregular thickness and/or dissimilar resistance to deformation, whereby adjustment is made according to the equation in dependence of continuous measurement of the roll pressure, the symbols in the equation designating the following:

h=roll gap opening (thickness of exiting rolled stock) F=roll force c=spring constant of the stand, and

S0=size of the roll gap without rolling material It is a well known fact that mill housings under load are more or less subject to stretching and that an entirely rigid system with a constant roll gap cannot be obtained. Changes in the thickness of the material entering the roll gap bring about an increase or decrease of the roll force and, therefore, of the roll gap andof the thickness of the rolled stock as it leaves the mill. In order to keep the springiness of the stand and the rolled stock tolerances resulting therefrom within suitable low limits, the stands and other built-in parts for the mill housings have been designed with large sections, although smaller sctions would do as well. With conventional rolling mill constructions, the portion of the expansion due to the housing itself amounts to about 20-50% of the total, because the screwdowns and chocks as well as the rolls under the rolling load are likewise subject to elastic deformation. Because of this fact, the result is that, even with the stands prestressed beyond the maximum possible rolling load, changes in the thickness of the exiting rolled stock can be only partially balanced out.

The increasing need for rolled stock with tolerances on thickness has led to the development of special thickness-regulating systems. These thickness-regulating systems make use of a change in the effective roll gap through the mechanical roll adjustment in response to roll force or thickness fluctuations. The drawback encountered with these thickness-regulating systems resides in the fact that, by reason of the large masses and considerable fn'ctional forces, the adjustment is extremely sluggish. Therefore, readjustment of the roll gap takes place in most cases too late, because of present mill speeds, so that an effective reduction of the thickness tolerances is, in most cases, unattainable.

The object of the present invention is to obviate the drawbacks inherent in the prior art thickness-regulating systems through the utilization of simple technical means; consequently, the invention is based on the problem of finding a method and apparatus for the production of strip, bars or the like of constant thickness, especially those of steel or other metal, by means of which the thickness tolerances in the rolled stock can be considerably reduced. The solution of this problem from an engineering point of view lies in accordance with the invention in the fact that a preliminary prestressing of the Patented Apr. 25, 1967 housing, preferably from the inside toward the outside, is brought about and altered in accordance with roll force fluctuations in inverse proportion to the latter. The invention is especially useful if the mill housing is prestressed hydraulically. It is well known in the art to employ a mill housing in which, prior to the rolling, the stands are hydraulically stretched through special pressure rods and wherein, during the rolling, the hydraulic pressure is diminished, in order to use the elastic force stored in the stands to exert an additional roll force upon the rolled stock. Aside from the fact that the use of hydraulically adjustable press rods exclusively for the stretching of the stands presents a not-unsubstantial additional expenditure and requires greater space for the roll housings, it is also true that optimum thickness regulation through the use of this arrangement could not be obtained, because only the springiness of the housing columns is taken into consideration. In accordance with the invention, it is proposed not only to regulate the thickness by use of the elastic column expansion, but also by use of the spring action of the screw-downs and the chocks; furthermore, the roll-force-dependent roll flattening and roll fiexure is indirectly evaluated for thickness regulating purposes.

In a further development of the method and apparatus, in accordance with the invention, it is contemplated'that the preliminary stressing of the stand be changed in accordance with the total load Q=F+P in accordance with the function wherein in the equation F=roll force,

P=prestressing of stand,

C =spring constant of the housing, and C =spring constant of the housing and rolls.

Furthermore, in connection with the invention, the prestressing force is set higher than any possible roll force increase, and the latter is automatically changed in accordance with roll force fluctuations. If the method of the invention is employed in a mill employing hydraulic means for balancing the back-up rolls, there is a special advantage in the fact that the hydraulic back-up roll balancing pressure can be utilized and increased severalfold, and the elastic expansion of the stand can be changed independently of the roll force. For this purpose, it will usually suffice to set the hydraulic balancing pressure for an increase of two or four times the normal value.

A sturdy device to carry out the method, which in its setup requires a minimum of expenditure and very little mounting space, is characterized by the fact that the prestress members are arranged between the top and bottom back-up roll chocks of the roll housings. If high-pressure hydraulic cylinders are used as prestressing members, they can serve simultaneously for the back-up roll balancing and the result is a directly-responding and accurate strip thickness regulation. A further feature of the device, according to the invention, consists in the fact that the high-pressure hydraulic cylinders are connected to a hydraulic unit, by means of which the occasional prestressing of the stand can be determined. It is also a feature of the invention that, as a control means for actuating the prestressing members, pressure dynamometers are provided. They are preferably positioned between the bottom back-up roll chock and the bottom cross-bar of the housing.

Finally, between the top and bottom work roll chocks, are arranged supporting members with slight supporting pressure which maintain the work rolls in contact with 3 the back-up rolls. These supporting members are also preferably hydraulic cylinders;

The invention will now be described in greater detail and reference will be made to the following drawings in which:

FIG. 1 is a schematic view of a roll housing in side elevation equipped'with a thicknessregulating system constructed in accordance with the principles of the invention; v

FIG. 2 is a vertical sectional view of the roll housing in FIG- 1' in which the'left-hand column is a vertical section through the roll axes and the right-hand column 7 is a section taken along the line II of FIG. 1,

FIG. 3 is a diagram showing graphically the changes of the roll force resulting from the dissimilar thickness of I entering rolled stock during the rolling process and also changes in the thickness' of the rolled material leaving the mill as the sum'of roll gap and elastic expansion of the stand for afnormal roll housing,

FIG. 4 is a diagram reproducing graphically the corresponding conditions for a roll housing put under a prestressing pressure, according to the invention,

, FIGVS 'is a schematic representation ofthe setup of a simplified design of a roll housing working with 1313-; a

stressed members, according to the invention,

FIG. 6 is a diagram showing the practical effect of the thickness regulating method according to the invention,

taking into consideration r-oll flattening and roll flexure,

and t V V 4, FIG. 7 is a diagram revealing the operating method of a mo'difiied'system for regulating the thickness of rolled stock. a

in which hroll gap opening,

F :roll force,

c=spring constant of the stand, and S=size of the roll gap without rolled stock.

The operator always tries for a constant thickness h V V sary, with the increasing rollrforce F resulting from the the augmented thickness of the entering rolled piece, for the roll gap S0 to be' decreased accordingly. Withnthe roll'force F decreasing as a result of the reduced thickness of the entering rolled stock, the roll gap SO mus be increased accordingly. 7 V

In FIGS 1 and 2 of the drawings, there is illustrated V a four-high rolling mill designed for the production of is wide strip, the mill having two housing columns 1 and 2 joined together at the top and bottom by cross-bars 3 'and 4. a The four-highlmill is provided with a top back-r up roll 5 and a bottom back-up roll 6wh ich have rela-t tively large diameters, and with a top work roll 7 and a 'bottorn'work roll' 8, each of relatively smaller diameter.

The two necks 5a and 5b of the top back-up rollr5 are each movably mounted in aso-called back-up roll chock 9 and 10; In' the same manner, the two necks 'fia and 6b ;of the bottomrback -up roll 6 are pivotally mounted a each in a back-up roll'chock 11 and 12. The backup roll'chock 9 of the top back-up roll 5 and the back-up roll chock 11 of the bottom back-up roll 6 are guided for vertical sliding movement in a window 13 built into the housing column'l, whereasjthe back-up roll chock 10 of the top aback-up roll 5 and the back-up roll chock 12 of the bottom back-up roll '6 are guided for vertical "sliding movement in a common window 14 of the housing column 2," The necks 7a and 7b of the top work roll 7 are positioned in work roll chocks 15 and 16, while the necks 8a and 8b of the bottom work roll 8 are seated in work roll 'chocks 17 and :18. Cutouts 19 and 2 0 in the two top back-up roll chocks 9 and 10 receive the two work roll chocks 15 and 16 of the top work roll 7 for 7 vertical sliding movement and corresponding cutouts 21,

22 in the bottom back-up roll'chocks 11 and 12 serve for 5 'the vertical guidance of the two work roll chocks 17 and 18 of the bottomwork roll 8.

Associated with each one of the housing columns 1 v and 2 is a screw down system consisting of an adjusting screw 23 actuated by an adjusting motor 25 through a gear 24. By means of the screwdown systems, whose adjusting screws 23 act against the top back-up roll delivered from a mill stand is not only dependent on the size of the roll gap which is present, but it is determined also by several single factors in the equation Using the equation 7 th=so+ as a basis, the result, according to FIG. 3, is that, in the case of an entering rolled stock thickness 'ho with the associated roll force F and the roll'gap S0, at rolled stock thickness h'will result. With a decrease in the entering rolled stock thickness from ho' to ho with constant roll gap SO, the result will be of necessity a lower roll form F and therewith a finished rolled stock thickness hi. The values will change accordingly with a'growth in the entering rolled stock thickness from'ho to ho son that the roll force rises to F and the exiting rolled stock thickness to I1 In'order to bring'the finish roll'stock thicknesses h and hi to the desired value h, despite the, W 1 inevitably variable thicknesstmeasurements, of the ente'r- 7 ing rolled stock and despiteJthe fluctuating will force, it has been suggested by the prior art to c ausethe roll gap SO to change bymeans of screw-down devicesfto SQ or S0 whereby the roll force drops down further from F to F or'increases further from F to-F This type of regulation aims at a constant exiting strip thickness 1:,

r in that, according to the aforementioned equation, in

case of change of the term F/c, the other term SO must be equated accordingly. In order to meet these require 'ments, costly and complicated instruments will be needed,

' but in connection therewith there is a drawback due to 'chocks 9 and 10, the effective roll gap is set each time between the work rolls 7 and 8 backed up by the sup- 7 porting rolls 5 and 6. The thickness of the rolled piece the sluggishness of the mechanical adjustment through f the gears and screws of the screw-down devices. The

compensation of the roll gap SO by actuating the screw-,

down devicesis brought about in accordance with changes in the roll force by means of pressure-measuring me'm-f bers 27, such as pressure dynamometers. Through the readjustment of the roll gap SO inaccordance with the V changing roll pressure through the screw-down devices, there arises, especially with high rolling speeds, a substantial lag, so that the sluggishness of the mechanical adjusting devices manifests itself unfavorably.

Since, for effective regulation'of thickness, the requirement: 1 i

h 80 constant must always be met, but since the known methods of readjustment of the roll gap SO are too sluggish, the requirement is met, in accordance with the invention, by working with a modification of the prestressing of the rolling mill stand. Such prestressing of the stand need merely be adjustable within the range of roll-force fluc uations resulting from the changes in thickness of thc entering rolled material, in order to be able to correct the roll gap SO and the finished strip thickness at will and with practically no lag. According to the'emb odiga es? ment of the invention illustrated in FIGS. 1 and 2 of the drawings, it is possible to influence the elastic expansion of the stand by the application of forces from an electrohydraulic system. This system consists of high-pressure hydraulic cylinders 28 and 29 inserted between the backup roll checks 9 and 11 or and 12 and or rollpressure-operated switch members 27 which regulate the amount of cylinder feed. The elect-ro-hydraulic systems in the two housing columns 1 and 2 work together. Both hydraulic high-pressure piston-cylinder units 28, 29 of each one of the housing columns 1 and 2 are, for instance, connected to the high-pressure side of a pressure transmitter 30 on whose low-pressure side pressure fluid is presented by a storage tank 31 which, in turn, is maintained under adjustable storage pressure by a pressuregas container 32. Adjustment of the pressure-gas pressure in the storage is provided by regulating members 33, 34, such as servo valves, which are actuated by the pressure dynamometers.

It is especially useful and inexpensive to use the wellknow hydraulic back-up roll balancing system to regulate the strip thickness. In connection therewith, it will sufiice in most cases to set the working pressure in the back-up roll balancing system about four times higher than the normal case of exclusive back-up roll balancing. This is done in order to change, even with an unfavorable rolling schedule, through modification of the prestress pressure (balancing-out pressure), the elastic expansion of the stand and therewith the roll gap SO, that thickness deviations of :10% in the first pass and i5% in the final pass can be handled. Normal gauge variations in cold strip only amount to :8 to 3%. FIGS. 1 and 2 show clearly that the high-pressure hydraulic cylinders 28, 29 through the back-up roll chocks 9, 11 or 13, 12, operating by means of the adjusting screws 23, on the one hand, and the pressure-measuring members 27 on the other hand, transmit their working pressure to the two housing columns 1 and 2. By this means, the entire elastic expansion of the stand and, indirectly, the rollsforce-dependent roll flattening and roll flexure, can be used to correct the roll gap SO. Thus, if a medium hydraulic prestressing pressure is used, both the expansion of the stand and the roll gap SO are increased accordingly. Lowering of the pressure, of course, will bring an opposite modification. The roll flexure will not be directly influenced by the prestressing cylinders 28 and 29, because no additional bending moments can occur at the roll necks. The utilization of pressure dynamometers 27 for the rollpressure-regulation of the prestressing of the stand is, therefore, especially advantageous, because, with their help, lags in the regulation are avoided. The pressuremeasuring dynamometers always indicate the sum Q of the roll force F, the weight G of the rolls and chocks, and the prestressing force P of the stand.

Accordingly, as the result of the arrival at the stand of a heavier rolled stock gauge h the roll force F, the elastic expansion of the stand P F We and the finished rolled stock gauge h will increase; also, the total load Q=F+G+P felt by the pressure dynamometer will become greater. Because the weight G of the roll is constant, the pressure exerted upon the cylinders 28, 29 depends on the control value available at the dynamometer, this value being reduced to a degree where the prestress pressure of the stand is low enough so that the finished rolled stock thickness h is again obtained. In other words, a reduction of the medium prestress. pressure P of the stand results in a lowering of the elastic expansion P/ C of the stand and, therefore, of the roll gap SO and of the thickness of the finished rolled stock, so that the desired value is again attained.

The diagram of FIG. 4 indicates that, with a lowering 6 of the entering rolled stock thickness from h to h with constant roll gap SO, there occurs a lower roll force F and a total pressure F i-P, which is smaller than F +1, as well as a finished rolled stock thickness h If one now increases the average prestress pressure P by AP; to P the roll gap S0 is simultaneously increased to S0 The total pressure F +P rises again to the rated value F +P ='-+P. The thickness of the rolled stock leaving the stand is then changed from h to the desired thickness h. Accordingly, the values change wlth an increase in thickness of the entering rolled stock from he to I20 in such a manner that the roll force rises to F the sum pressure to F -l-P, and the exitmg rolled stock thickness to 11;. However, by lowering the medium prestress pressure by AP from P to P the roll gap S0 is reduced to S0 and the total pressure drops again from F +P to the desired value F +P =F+P. The finished I rolled stock thickness is, at the same time, reduced from 11 to the rated value h. In both cases, the prestress force P is reduced to a degree, until the total pressure F +P is again attained, whereby, at the same time, the finished rolled stock thickness returns automatically to the desired value h. It is a condition, therefore that the deviations from the total pressure F +P (as ind1cated by the pressure measuring means) are regulated through modification of the expansion P of the stand, wherewith the desired thickness h is attained automatically.

With this method, the mechanical screwdown need not be actuated and determination of the prevailing roll gap S0 is not necessary. Costly and complicated regulating instruments are not needed, because the apparatus for 1nfluencing the expansion of the stand is immediately and directly effective in regulating the roll gap. It is also possible to perform the adjustment separately for each side of the housing. The columns 1 and 2 can be built considerably lighter, because their expansion is rendered ineitectual by the prestressing of the stand; its absolute s1ze has no influence upon the rolled stock tolerances. The actual size of the column cross-sections are selected, therefore, in accordance with the permissible load alone as a factor. A further important advantage of this method consists in the possibility of the installation of the present thickness-regulating system in many existing rolling mills and its application to all types of rolling, i.e., both cold and hot rolling, with a minimum of expenditure.

In accordance with the invention, it is also possible to obtain a fairly constant thickness of rolled material without automatic gauge regulation. FIG. 5 indicates that the screwdown pressure Q is composed theoretically of the prestress pressure P and the roll force F (Q=P+F). The load Q (screw-down pressure) is taken up by the supports P and P as well as F by the rolls and stock. If the support F takes over a larger portion of the load, the supports P and P are thereby unburdened without the total load Q being changed. On the other hand, with a lowering and unburdening of the support F, a greater absorption of the load will ensue for the supports P -l-P If it is assumed that the support F is formed by the rolled stock with its roll force and that P and P are prestress forces between the bottom and top back-up roll chocks, then, in case of a modification of the roll force F by reason of changes in the thickness of the entering rolled stock, the support force P=P +P (prestress force) becomes adapted to the changes. It is a condition, however, that the prestress force P=P +P be greater than any possible roll force increase through an increment in the thickness of the entering rolled stock. In such a case, a case, a twoto threefold increase of the customary backup roll balancing pressure would make possible approximate regulation of the fluctuations in thickness of the entering rolled material. Only when the growing roll force AF becomes greater than the prestress force P will there be an increase in the screw-down pressure Q and, therefore, in the elastic expansion of the stand. The roll gap SO will become greater as a consequence thereof, -so that V justing movement as it occurs.

the supports P and P however, must not change after 7 the gauge of the finished rolled material will become heavier.

In practice, the prestress force P, during the operation of the screw-downs, should always be maintained at a constant value and should follow the adjusting movement. This can be achieved by connecting the hydraulic back-up roll balancing equipment to a weight accumulator, which will always cause the prestress cylinders to follow the ad- The adjusted height of the completion of the adjusting movement; this can be accomplished by shutting off the connections to the balancing cylinders so that, with a change in the roll force F, the supports P and P can be burdened or unburdened accordingly. Theoretically, a constant finished rolled stock thickness h could 'be'maintained as long as the value of the roll force F remains lower than the prestress force P=P +Pg and as long as Q=P+F remains constant. In practice, it is possible through the prestress force P'to compensate only for the elastic expansion of the stand,

i.e., theelastic expansion of the columns 1, 2, of the screw-downs 23, and of the chocks 9 to 12, whereas the roll flattening and the roll flexure is dependent on the roll force and acts as a free spring member. Any change in the roll flattening and flexure results in a modification of the roll gap SO and, therefore, of the finished rolled stock thickness. The influence of roll flattening and roll flexure j upon'the thickness of the rolled stock, however, is slighter than the remaining elastic expansion of the mill housing, so that deviations from the desired thickness of the rolled stock can be kept within limits.

The practical effect of the 'predescribe'd gauge regulating method in conjunction with roll flattening and roll flexure are revealed in FIG. 6. With a reduction of the entering rolled stock thickness from ho to I the roll force will normally drop from F to F and the thickness 0f the rolled stock leaving the stand from h to I2 In the extreme case, .at I1 the roll flattening, roll flexure and roll force would be equal to 0, and the entering gauge would be equal to the finished gauge. The support force P between the back-up roll chocks would, in this ica'se, b e equal to the adjusting force Q, since (according to the equation Q=P+F)' the sum of termbecomes F =0.

With an increase in the thickness of the incoming rolled stock from ho to ho the roll force would normally rise from F'to F and the-finished rolled stock thickness from h' to hg. By applying'the method with constant prestress value (support) between the back-up roll chocks, only the roll flattening and the roll fiexure and, therefore, theroll gap SO can increase by reasonof the rising roll force F. The roll force will then grow from F to F and the exiting rolled stock thicknessfrom h to 12 If the roll force F, by reason of great increasein thickness of the entering rolled stock rises to a degree where F becomes greater than the support force} P, a lifting of the back-up roll chocks oil the fixed supports occurs and the supportforce ;P becomes equal to 0. The screw-down pressure Q is, in this case, equal to the roll force F according to' the equation Q=F, because the term P drops out. After the lifting of the'back-up roll chocks,'the screw-down pressure Q increases by'Q=FP and the spring characteristic of the entirestand thereupon again becomes effective.

' stock in accordance with a relatively flat spring characteristic of the entire stand.

This method also has the advantage that the gauge tolerances of the rolled material can be kept to a minimum without actuating the mechanical screw-downs and withoutthe necessity of providing costly and complicated regulating instruments. Also, the main columns of the.

mill stand could be built lighter, because any expansion support, so that the stand expansion no longer has any influence upon the gauge of the rolled stock. Consequently, the sections of the columns can be designed in accordance with load only. Here, too, subsequent installation of the support and stand prestressing device is possible in any existing rolling mill and is applicable to all types of rolling.

The application of the above-described concepts is also possible in combination; according to the last-described method with fixed supports between the back-up roll chocks there occurs a far-reaching automatic correction of thickness deviation and, then, further compensation'is brought about by use of the first-described method' As the precise regulation ing to the first-mentioned method.

port force P would have to be reduced further 'from P to P and with decreasing thickness of the entering rolled material the changing support force P would have to be increased further from P to P In both cases, the

expected thicknesses I1 and I1 of the finished rolled' stock are thereby maintained at the desired'thickness h.

In accordance with the provisions of the patent stat utes, the principle and operation of the invention have been illustrated and described in what is considered to representthe best embodiment thereof. However, it is desired to have it understood that, within the scope of the appended claims, the invention may be practiced otherwise than as specifically illustrated and described.

I claim: a 7,

1. In a rolling mill including a housing for processing elongated material such as strip, bars and the like wherein the material is characterized by portions offering irregucomprising:

means for measuring the rolling force developed during processing, means .for varying the elastic expansion of the housing pursuant to a change in the rolling force to maintain the thickness of the processed material substantially uniform according to the equation wherein:

h=the thickness of processed material,

P=prestressing of the housing,

F =rolling force,

C =spring constant of the housing,

C zspring constant of the housing including the rolls, and

SO=size of the roll gap when no material is being rolled.

2. In a rolling mill according to claim 1 including:

a fluid pressure member for prestressing the mill, and

control means for varying the pressure of said fluid pressure members.

3. In a rolling mill according to claim 1 wherein said i V mill is a multi-high .mill including a pair of Work and back-up rolls and having checks for the back-up rolls thereof, and

includes prestressing members. arranged between the top and bottom back-up roll chocks for varying the elastic expansion of the mill.

4. In a rolling mill according to claim 3 including control means for varying the pressure in said prestressing members.

5. In a rolling mill according to claim'3 wherein said prestressing members are high pressure hydraulic'cylinders that serve also for balancing of the back-up rolls. 6. In a rolling mill according to claim 5 including a weight accumulator connected to said high pressure cylinders.

7. In a rolling mill according to claim 1 wherein said measuring means is a pressure dynamometer.

8. In a rolling mill according to claim 3 wherein said measuring means comprise dynamometers located between the bottom back-up roll chocks and housing.

9. In a rolling mill according to claim 3 including pressure supporting means arranged between the top and bottom work roll chocks for urging the work rolls into contact with the back-up rolls.

10. In a rolling mill according to claim 9 wherein said pressure supporting members comprise hydraulic cylinders.

11. A rolling mill, comprising (a) a pair of frames, each having a window,

(b) at least a pair of chocks slidable in each window,

(c) at least a pair of rolls rotatably mounted in the chocks,

(d) a screwdown associated with each frame and operative to compress the chocks and rolls toward each other,

(e) a prestressing device associated with each frame and operative to separate each pair of chocks and to place the respective frame under an initial stretching force,

(f) a control measuring the rolling force as material passes between the rolls and operating the said devices to regulate the stretching forces so that the product of the stretching forces and the spring constants of the frames is maintained equal to the prodnot of the rolling force and the spring constant of the rolls, screwdown, and frames.

References Cited by the Examiner UNITED STATES PATENTS CHARLES W. LANHAM, Primary Examiner.

20 R. D. GREFE, Assistant Examiner. 

1. IN A ROLLING MILL INCLUDING A HOUSING FOR PROCESSING ELONGATED MATERIAL SUCH AS STRIP, BARS AND THE LIKE WHEREIN THE MATERIAL IS CHARACTERIZED BY PORTIONS OFFERING IRREGULAR RESISTANCE TO DEFORMATION SUCH AS DIFFERENT THICKNESSES COMPRISING: MEANS FOR MEASURING THE ROLLING FORCE DEVELOPED DURING PROCESSING, MEANS FOR VARYING THE ELASTIC EXPANSION OF THE HOUSING PURSUANT TO A CHANGE IN THE ROLLING FORCE TO MAINTAIN THE THICKNESS OF THE PROCESSED MATERIAL SUBSTANTIALLY UNIFORM ACCORDING TO THE EQUATION 